CAPS 2009 Problem 5

The $n$ -tuple $(a_1,a_2,\ldots,a_n)$ of integers satisfies the following: list $1\le a_1<a_2<\cdots < a_n\le 50$ (ii) for each $n$ -tuple $(b_1,b_2,\ldots,b_n)$ of positive integers, there exist a positive integer $m$ and an $n$ -tuple $(c_1,c_2,\ldots,c_n)$ of positive integers such that $$mb_i=c_i^{a_i} \text{for } i=1,2,\ldots,n.$$ [/list]Prove that $n\le 16$ and determine the number of $n$ -tuples $(a_1,a_2,\ldots,a_n$ ) satisfying these conditions for $n=16$ .